An INTRODUCTION TO HIDDEN MARKOV MODEL By: Pejman Golshan Shu Yu

HIDDEN MARKOV MODEL(HMM) Real-world has structures and processes which have observable outputs. – Usually sequential . – Cannot see the event producing the output. Problem: how to construct a model of the structure or process given only observations ?

HISTORY OF HMM • Basic theory developed and published in 1960s and 70s • No widespread understanding and application until late 80s • Why? – Theory published in mathematic journals which were not widely read. – Insufficient tutorial material for readers to understand and apply concepts.

Andrei Andreyevich Markov 1856-1922 Andrey Andreyevich Markov was a Russian mathematician. He is best known for his work on stochastic processes. A primary subject of his research later became known as Markov chains and Markov processes .

HIDDEN MARKOV MODEL • A Hidden Markov Model (HMM) is a statical model in which the system is being modeled is assumed to be a Markov process with hidden states. • Markov chain property: probability of each subsequent state depends only on what was the previous state.

HMM COMPONENTS • A set of states (x’s) • A set of possible output symbols (y’s) • A state transition matrix (a’s) – probability of making transition from one state to the next • Output emission matrix (b’s) – probability of a emitting/observing a symbol at a particular state • Initial probability vector – probability of starting at a particular state – Not shown, sometimes assumed to be 1

The Example we had in the class Two states : ‘Rain’ and ‘Dry’. Transition probabilities: P(‘Rain’|‘Rain’)=0.3 , P(‘Dry’|‘Rain’)=0.7 , P(‘Rain’)=0.6 . P (‘rain’|‘Dry’)=0.2, P(‘Dry’|‘Dry’)=0.8 Initial probabilities: say P(‘Rain’)=0.4 , P(‘Dry’)=0.6

CALCULATION OF HMM By Markov chain property, probability of the state sequence can be found by: Suppose we want to calculate a probability of a sequence of states in the example, {Dry,Dry, Rain,Rain}

PROBLEMS OF HMM • Three problems must be solved for HMMs to be useful in real-world applications Evaluation: • Problem - Compute Probability of observation sequence given a model • Solution - Forward Algorithm and Viterbi Algorithm Decoding: • Problem - Find state sequence which maximizes probability of Observation sequence • Solution - Viterbi Algorithm Training: • Problem - Adjust model parameters to maximize probability of observed sequences • Solution - Forward-Backward Algorithm

EVOLUTION OF PROBLEM Given a set of HMMs, which is the one most likely to have produced the observation sequence?

DECODING PROBLEM

Applications • It is able to handle new data robustly Benefit of Hidden Markov Model • It is able to handle new data robustly • Computationally efficient to develop and evaluate (due to the existence of established training algorithms). • It is able to predict similar patterns efficiently

Applications Stock prediction

Applications Stock prediction Stock behavior of past is similar to behavior of current day The next day’s stock price should follow about the same past pattern Using trained HMM likelihood value P for current day’s dataset can be calculated. from the past dataset we can locate those instances that would produce the nearest P likelihood value.

Applications Speech recognition Observation: speech waveform phone model (phonetic symbols) word model

Applications Face Expression Characterization – Using HMM framework The hidden state of HMMs is the hidden emotional state of the individual. The observable symbols of HMMs is the feature vectors extracted from face videos The State Transition matrix and Observation probability matrix of HMMs is Dynamical information extracted from videos accompanied by observation symbols extracted using VQ(vector Quantization)

Applications Computational finance Alignment of bio-sequences Single Molecule Kinetic analysis Time Series Analysis Cryptanalysis Activity recognition Speech recognition Protein folding Speech synthesis Metamorphic Virus Detection Part-of-speech tagging DNA Motif Discovery Document Separation in scanning solutions Machine translation Partial discharge Gene prediction Handwriting Recognition

References Nguyen, N and Nguyen, D; Hidden Markov Model for Stock Selection, Journal of Risks in special issue: Recent Advances in Mathematical Modeling of the Financial Markets, Risks 2015, 3(4), 455-473; doi:10.3390/risks3040455. Cs.cmu.edu. (n.d.). Cite a Website - Cite This For Me. [online] Available at: http://www.cs.cmu.edu/~bdhingra/papers/stock_hmm.pdf. . wordspotting, H. (n.d.). HMMs in Speech recognition and wordspotting. [online] Kapelnick.blogspot.ca. Available at: http://kapelnick.blogspot.ca/p/blog-page_14.html. University of Illinois, C. (n.d.). Microsoft power point Face recognition. [online] Slideshare.net. Available at: https://www.slideshare.net/aadishchopra/microsoft-power-point-face-recognition.

Thank you!